Correct reasoning in logic. Right and wrong reasoning

Here are two examples of deductive conclusions from the story of the Russian humorist of the beginning of the century V. Bilibin.

“If the sun did not exist in the world, then we would have to constantly burn candles and kerosene.

If you had to constantly burn candles and kerosene, then the officials would not have enough of their salaries and they would take bribes.

Therefore, officials do not take bribes because the sun exists in the world.

“If bulls and chickens went fried, then there would be no need to build stoves and, therefore, there would be fewer fires.

If there were fewer fires, the insurance companies would not raise insurance premiums so cruelly.

Therefore, the insurance companies raised the insurance premium so cruelly because bulls and chickens do not go fried.

These arguments parodied the usual once-naive explanations of why officials take bribes and insurance companies inflate insurance rates.

It is clear that both of these arguments are logically untenable. Their conclusions do not follow from the premises they have accepted. Therefore, even if the premises were true, this would not mean that the conclusions are true.

The main task of logic is to separate right ways reasoning (conclusion, inference) from the wrong ones. Correct conclusions are also called justified or logical.

The peculiarity of formal logic in the approach to the analysis of the correctness of reasoning is associated with its basic principle, according to which the correctness of reasoning depends only on its form, or scheme. by the most in a general way the form of reasoning can be defined as a way of connecting the substantive parts included in it.

In correct reasoning, the conclusion follows from the premises with logical necessity, and the general scheme of such reasoning is a logical law.

Logical laws thus underlie logically perfect thinking, constituting that invisible iron frame on which any consistent reasoning rests. To reason logically correctly means to reason in accordance with the laws of logic. This explains the importance of these laws.

There are an infinite number of schemes for correct reasoning (logical laws). Many of them are known to us from the practice of reasoning. We apply them intuitively, without realizing that in each correctly drawn conclusion we use one or another logical law.

Here are some of the most commonly used schemes.

“If there is a first, then there is a second; there is the first, therefore there is the second. This scheme allows us to pass from the statement of the conditional statement and the statement of its foundation to the statement of the consequence. For a logically correct transition, the specific content of the premises and conclusion does not matter, only the way they are connected is important. Therefore, in the scheme, instead of statements with a certain content, “meaningless” phrases “there is the first” and “there is the second” are used. According to the scheme under consideration, in particular, the reasoning proceeds: “If the ice heats up, it melts; ice is heated; it means he's melting."

This logically correct movement of thought is sometimes confused with its similar but logically incorrect movement from asserting the consequence of a conditional statement to asserting its foundation: “if there is a first, then there is a second, there is a second; so there is a first. Last scheme is not a logical law, from true premises it can lead to a false conclusion. Let's say that the reasoning following this scheme is “If a person has a fever, he is sick; the person is sick; therefore, he has a fever” leads to the erroneous conclusion that the disease always proceeds with an increase in temperature.

“If there is a first, then there is a second; but there is no second; so there is no first. By means of this scheme, from the affirmation of a conditional statement and the negation of its consequence, a transition is made to the negation of the foundation of the statement. For example: “If the day comes, it becomes light; but now it is not light; therefore the day has not come.” Sometimes this scheme is confused with the logically incorrect movement of thought from denying the basis of the conditional statement to denying its consequence: “if there is the first, then there is the second; but the first is not; therefore, there is no second "(" If a person has a fever, he is sick; but he does not have a fever; therefore, he is not sick ").

Returning to the two arguments about officials who do not take bribes because the sun is shining, and about insurance companies that inflate the insurance percentage because bulls and chickens do not go fried, we can note that this incorrect scheme underlies these arguments.

"If the first leads to the second, then if the second leads to the third, then the first leads to the third." This scheme, which at first glance seems cumbersome, is often and without difficulty used in a wide variety of arguments. For example: “If it is the case that with the growth of knowledge about a person, the ability to protect him from diseases increases, then if with an increase in this ability the average duration of human life increases, then with the growth of knowledge about a person, the average duration of his life increases.”

“If there is a first, then there is a second; therefore, if there is no second, then there is no first. This scheme allows, using negation, to swap statements. For example, from the statement “If there is an effect, there is also a cause”, the statement “If there is no cause, there is no effect” is obtained.

“There is at least the first or the second; but the first is not; so there is a second one. For example: “It happens day or night; now there is no night; therefore, it is now day."

“Either the first or the second takes place; there is the first; so there is no second. By means of this scheme, from the assertion of two mutually exclusive alternatives and the establishment of which of them is available, the transition is made to the denial of the other alternative. For example: “Dostoevsky was born either in Moscow or in St. Petersburg; he was born in Moscow; so it is not true that he was born in St. Petersburg. In the American western The Good, the Bad and the Ugly, one can hear the following magnificent division of human roles. The bandit says: “Remember, One-armed, that the world is divided into two parts: those who hold a revolver, and those who dig. I have a revolver now, so take a shovel. This reasoning is also based on the scheme under consideration.

“It is not true that there is both the first and the second; therefore, there is no first or no second”; “there is the first or there is the second; therefore, it is not true that there is no first and no second. These and similar schemes allow you to move from statements with the union “and” to statements with the union “or”, and vice versa. Using these diagrams, one can go from the statement “It is not true that the study of logic is difficult and useless” to the statement “The study of logic is not difficult or it is not useless” and from the statement “Amundsen or Scott was the first at the South Pole” to the statement “False that neither Amundsen nor Scott is the first person to visit the South Pole.

These are some of the infinite number of schemes of correct reasoning at our disposal.

CHARACTERISTIC ERROR

Usually we apply logical laws without thinking about them, often unaware of their very existence. But it happens that the use of even a simple scheme faces certain difficulties.

Experiments conducted by psychologists in order to compare the thinking of people of different cultures clearly show that most often the reason for the difficulties is that the reasoning scheme, its form is not distinguished in pure form. To resolve the question of the correctness of the reasoning, some irrelevant substantive considerations are instead involved. Usually they are associated with a specific situation described in the argument.

This is how M. Cole and S. Scribner describe the course of one of the experiments conducted in Africa in the book “Culture and Thinking”.

Experimenter.

One day the spider went to a festive dinner. But he was told that before he began to eat, he had to answer one question. The question is: “The spider and the black deer always eat together. The spider is eating. Does the deer eat?

Subject. Were they in the forest?

Experimenter. Yes.

Subject. Did they eat together?

Experimenter. Spider and deer always eat together. The spider is eating. Do deer eat?

Subject. But I wasn't there. How can I answer such a question?

Experimenter. Can't answer? Even if you weren't there, you can answer this question. (Repeat question.)

Subject. Yes, yes, black deer eats.

Experimenter. Why are you saying. what does black deer eat?

Subject. Because the black deer always walks through the forest all day and eats green leaves. Then he rests for a while and gets up again to eat.

There is an obvious mistake here. The subject does not have a general idea of ​​the logical correctness of the conclusion. To give an answer, he seeks to rely on some facts, and when the experimenter refuses to help him in search of such facts, he himself invents them.

Another example from the same study.

Experimenter. If Flumo or Yakpalo drink cane juice, the village chief becomes angry. Flumo does not drink cane juice. Yakpalo drinks cane juice. Is the village chief angry?

Subject. People don't get angry at other people.

The experimenter repeats the task.

Subject. The village chief was not angry that day.

Experimenter. Wasn't the village chief angry? Why?

Subject. Because he doesn't like Flumo.

Experimenter. He doesn't like Flumo? Tell me why?

Subject. Because when Flumo drinks cane juice, it's bad. That's why the village chief gets angry when Flumo does this. And when Yakpalo sometimes drinks cane juice, he does nothing bad to people. He goes and goes to bed. So people don't get angry with him. But those who get drunk on cane juice and start to fight - the headman cannot tolerate them in the village.

The subject most likely has in mind some specific people or simply invented them. He discarded the first premise of the problem and replaced it with another statement: people don't get angry with other people. Then he introduced new data into the problem concerning the behavior of Flumo and Yakpalo. The subject's answer to the experimental problem was incorrect. But it was the result of quite logical reasoning based on new premises.

To analyze the problem posed in the first experiment, we reformulate it so that the logical connections of the statements are revealed: “If a spider eats, then a deer also eats; if the deer eats, then the spider also eats; spider eats; therefore the deer also eats.” There are three messages here. Does the two of them: "If the spider eats, the deer also eats" and "The spider eats" follow the conclusion "The deer eats"? Certainly. The reasoning goes according to the already mentioned scheme: “if there is the first, then there is the second; there is the first; so there is a second one. It is a logical law. The correctness of this reasoning does not, of course, depend on whether everything happens in the forest, whether the subject was present, etc.

The scheme used in the second problem is somewhat more complicated: “If Flumo or Yakpalo drink cane juice, the headman of the village gets angry. Flumo does not drink cane juice. Yakpalo drinks cane juice. Is the village chief angry?” Abstracting from the specific content, we reveal the reasoning scheme: “if there is the first or the second, then there is the third; there is no first, but there is a second; therefore, there is a third. This scheme is a logical law, which means that the reasoning is correct. The scheme is close to the previously mentioned scheme “if there is a first, then there is a second; there is the first; therefore, there is a second. The only difference is that two alternatives are indicated as the "first" in a more complex argument, one of which is immediately excluded.

CONVINCING REASONS

"Fearing ... your own shadow and your own ignorance, do not part with a reliable and true foundation."

"Scientific evidence should not be required of a speaker, just as emotional persuasion should not be required of a mathematician."

Aristotle

"Evidence is valued for quality, not quantity."

Latin proverb

“The arguments that a person comes up with on his own usually convince him more than those that come to the mind of others.”

B. Pascal

“Only someone who knows nothing about cars will try to drive without gasoline; only he who does not understand anything in the mind will try to think without a solid, undeniable foundation.

Book: LOGIC FOR LAWYERS: LECTURES. / Law College LNU. Franco

4. Correct and incorrect reasoning. The concept of a logical error

In logic reasoning are divided into:

♦ correct;

♦ incorrect.

Correct reasoning- it is a reasoning that adheres to all the rules and laws of logic. Incorrect inference is a conclusion in which logical errors are made due to violation of the rules or laws of logic.

Logic errors are of two types:

♦ paralogisms;

♦ sophisms.

Paralogisms are logical errors that are made inadvertently (out of ignorance) in the processes of reasoning.

sophistry- these are logical errors that are deliberately made in reasoning processes with the aim of misleading the opponent, substantiating a false statement, some kind of nonsense, etc.

Sophistry has been known since ancient times. Sophists widely used such considerations in their practice. It is from them that the name comes. "sophism". Numerous examples of reasoning that the sophists used in various disputes have come down to our time. Let's take a look at some of them.

♦ The most famous ancient sophism is the reasoning, which was called "Horned".

Imagine a situation: one person wants to convince another that he has horns. The rationale for this is as follows: “What you have not lost, you have. You didn't lose your horn. So, at you have horns."

This reasoning seems correct at first glance. But there is a logical error in it, which a person who does not understand logic is unlikely to be able to immediately find.

♦ Let's take another example. Protagoras (the founder of the school of sophists) had a student Euathlus. The teacher and student made an agreement that Euathlus would only pay the tuition fees after he won his first lawsuit. But, having completed his studies, Evatl was in no hurry to appear in court. The teacher's patience snapped, and he filed a lawsuit against his student. "Evatl will have to pay me anyway, Protagoras argued. - He will either win this process or lose it. If he wins, he will pay by agreement; if he loses, he will pay according to the verdict of the court. "Nothing like this, Evatl objected. - Indeed, I either win the process or lose it.

If I win- a court decision will release me from the fee if same lose- I will not pay according to our agreement *.

This example also makes a logical error. And what exactly - we will find out further.

The main task of logic is the analysis of correct considerations. Logicians seek to identify and explore the schemes of such considerations, to determine their Various types etc. Incorrect reasoning in logic is analyzed only from the point of view of the errors that are made in them.

It should be noted that the correctness of the reasoning does not mean the truth of its assumptions and conclusions. In general, logic does not concern itself with determining the truth or falsity of the premises and conclusions of considerations. But in logic there is such a rule: if the reasoning is constructed correctly (according to the rules and laws of logic) and at the same time it is based on true premises, then the conclusion of such reasoning will always be unconditionally true. In other cases, the truth of the conclusion cannot be guaranteed.

So, if the conclusion is constructed incorrectly, then, even though its premises are true, the conclusion of such reasoning can be true in one case, and false in the second.

♦ As an example, consider the following two considerations, which follow the same wrong pattern.

(1) Logic is a science.

Alchemy is not logic.

Alchemy is not a science.

(2) Logic is a science.

Law is not logic.

Law is not a science.

Obviously, in the first argument, the conclusion is true, but in the second it is false, although the premises in both cases are true statements.

It is also impossible to guarantee the truth of the conclusion of the reasoning, when at least one of its founders is false, even if this reasoning is correct.

Correct inference - reasoning in which some thoughts (conclusions) necessarily follow from other judgments (premises).

An example of correct reasoning can be the following conclusion: “Every citizen of Ukraine must recognize its Constitution. All people's deputies Ukraine - citizens of Ukraine. Consequently, each of them must recognize the Constitution of their state”, and an example of a true thought is the judgment: “There are citizens of Ukraine who do not recognize at least some articles of the Constitution of their state.”

The following reasoning should be considered incorrect: "Since the economic crisis in Ukraine clearly makes itself felt after the proclamation of its independence, the latter is the cause of this crisis." This type of logical error is called "after this - because of this." It lies in the fact that the temporal sequence of events in such cases is identified with the causal one. An example of a false thought can be any provision that does not correspond to reality, for example, the assertion that the Ukrainian nation does not exist at all.

The purpose of knowledge is to obtain true knowledge. In order to obtain such knowledge with the help of reasoning, it is necessary, firstly, to have true premises, and secondly, to combine them correctly, to reason according to the laws of logic. When using false premises, they make factual errors, and when they violate the laws of logic, the rules for constructing reasoning, they make logical errors. Factual errors, of course, must be avoided, which is not always possible. As for the logical ones, a person of high intellectual culture can avoid these mistakes, since the basic laws of logically correct thinking, the rules for constructing reasoning, and even meaningfully typical mistakes in reasoning.

Logic teaches to reason correctly, not to make logical errors, to distinguish correct reasoning from incorrect. It classifies correct reasoning for the purpose of their systematic understanding. In this context, the question may arise: since there are many considerations, is it possible, in the words of Kozma Prutkov, to embrace the immensity? Yes, you can, because logic teaches you to reason, focusing not on the specific content of the thoughts that are part of the reasoning, but on the scheme, the structure of the reasoning, the form of combining these thoughts. Let's say a form of reasoning like "Every X is V, but the present G is X; therefore, given G is correct, and knowledge of its correctness contains much richer information than knowledge of the correctness of a separate meaningful reasoning of a similar form. And the form of reasoning according to the scheme "Each x is in and z also have V; hence, X There is X" refers to incorrect. Just as grammar studies the forms of words and their combinations in a sentence, abstracting from the specific content of linguistic expressions, so logic studies the forms of thoughts and their combinations, abstracting from the specific content of these thoughts.

To reveal the form of thought or reasoning, they must be formalized.

1.	LOGIC FOR LAWYERS: LECTURES. / Law College LNU. Franco
2.
3.	3. Historical stages in the development of logical knowledge: the logic of ancient India, the logic of ancient Greece
4.	4. Features of general or traditional (Aristotelian) logic.
5.	5. Features of symbolic or mathematical logic.
6.	6. Theoretical and practical logic.
7.	Topic 2: THINKING AND SPEECH 1. Thinking (reasoning): definition and features.
8.	2. Activity and thinking
9.	3. Structure of thinking
10.	4. Correct and incorrect reasoning. The concept of a logical error
11.	5. Logical form of reasoning
12.	6. Types and types of thinking.
13.	7. Features of the thinking of a lawyer
14.	8. Significance of logic for lawyers
15.	Topic 3: Semiotics as a science of signs. Language as a sign system. 1. Semiotics as the science of signs
16.	2. The concept of a sign. Types of signs
17.	3. Language as a sign system. language signs.
18.	4. The structure of the sign process. Sign value structure. Common Logic Errors
19.	5. Dimensions and levels of the sign process
20.	6. The language of law
21.	Section III. METHODOLOGICAL FUNCTION OF FORMAL LOGIC 1. Method and methodology.
22.	2. Logical methods of research (cognition)
23.	3. Formalization method
24.	BASIC FORMS AND LAWS OF ABSTRACT-LOGICAL THINKING 1. General characteristics of the concept as a form of thinking. Concept structure
25.	2. Types of concepts. Logical characteristics of concepts
26.	3. Types of relationships between concepts
27.	4. Operations with concepts 4.1. Limitation and generalization of concepts
28.	4.2. Concept division operation
29.	4.3. Addition, multiplication and subtraction of concepts (more precisely, their volumes)
30.	4.4 Concept definition operation
31.	BASIC FORMS AND LAWS OF ABSTRACT-LOGICAL THINKING II. Statements. 1. General characteristics of the statement
32.

Tasks of logic. 1. Correct reasoning. The word "logic" is used quite often, but in different meanings. Often they talk about the logic of events, the logic of character, etc. In these cases, they mean a certain sequence and dependence of events or actions, the presence of a certain common line in them. Formal logic is the science of the laws and operations of correct thinking. The main task of logic is to separate the correct ways of reasoning (conclusions, inferences)

from the wrong ones. Correct conclusions are also called justified, consistent or logical. Reasoning is a certain, internally conditioned connection of statements. Distinctive feature a correct conclusion is that it always leads from true premises to a true conclusion. 2. Logical form. The peculiarity of formal logic is connected, first of all, with its basic principle, according to which the correctness of reasoning depends only on its logical logic.

forms. In the most general way, the form of reasoning can be defined as a way of connecting the substantive parts included in this reasoning. 3. Deduction and induction. An inference is a logical operation, as a result of which a new statement is obtained from one or more accepted statements (premises) - a conclusion (consequence). Depending on whether there is a connection of a logical consequence between the premises and the conclusion, two types of inferences can be distinguished. In deductive reasoning, this connection is based on a logical

law, whereby the conclusion follows with logical necessity from the premises accepted. A distinctive feature of such an inference is that it always leads from true premises to a true conclusion. In inductive reasoning, the connection between premises and conclusions is not based on the law of logic, but on some factual or psychological grounds that do not have a purely formal character. In such a conclusion, the conclusion does not follow logically from the premises and may contain information that deviates

from them. Induction does not give a full guarantee of obtaining a new truth from the already existing ones. The maximum we can talk about is a certain degree of probability of the statement being deduced. Especially characteristic deductions are logical transitions from general to particular knowledge. 4. Intuitive logic. Intuitive logic is usually understood as intuitive ideas about the correctness of reasoning, which has developed spontaneously in the process of everyday practice of thinking.

Intuitive logic successfully copes with its tasks in Everyday life, but it is completely insufficient for criticizing incorrect reasoning. 5. Some schemes of correct reasoning. In correct reasoning, the conclusion follows from the premises with logical necessity, and the general scheme of such reasoning is a logical law. Logical laws underlie logically perfect thinking.

To reason logically correctly means to reason in accordance with the laws of logic. Here are some of the most commonly used schemes: If there is the first, then there is the second; there is the first; hence there is a second one. This scheme allows us to move from the assertion of a conditional statement and the assertion of its foundation to the assertion of a conditional consequence. If there is a first, then there is a second; but there is no second; so there is no first.

By means of this scheme, from the affirmation of a conditional statement and the negation of its consequence, a transition is made to the negation of the foundation of the statement. If there is a first, then there is a second; therefore, if there is no second, then there is no first. This scheme allows, using negation, to swap statements. There is at least either the first or the second; but the first is not; so there is a second one. For example: “There are days and nights; now there is no night; therefore, it is now day."

Either the first or the second takes place; there is the first; so there is no other. By means of this scheme, from the assertion of two mutually exclusive alternatives and the establishment of which of them is present, the transition is made to the negation of the other alternative. It is not true that there is both the first and the second; therefore, there is no first or second. Is there a first or is there a second; therefore, it is not true that there is no first and no second.

These and similar schemes allow you to move from statements with the union “and” to statements with the union “or”, and vice versa. 6. Traditional and modern logic. The history of logic covers about two and a half millennia. Only philosophy and mathematics are "older" than formal logic. In the first stage, usually called traditional logic, formal logic developed very slowly. Kant (1724-1804) said that formal logic is a complete science, not advanced

not one step since the time of Aristotle. G. Leibniz (1646-1716) gave a clear expression to the ideas to present the proof as a calculation, similar to the calculation in mathematics. Leibniz's ideas, however, did not have a noticeable influence on his contemporaries. Frege (1848-1925) in his work began to apply formal logic to the study of the foundations of mathematics. Frege was convinced that "arithmetic is part of logic and should not borrow from experience or contemplation.

no justification." The famous Russian physicist Ehrenfest was the first to put forward a hypothesis about the possibility of applying contemporary logic to technology. 7. Modern logic and other sciences. Since its inception, logic has been most closely associated with philosophy. For many centuries, logic was considered, like psychology, one of the "philosophical sciences". Mathematical logic arose, in essence, at the junction of two such different sciences as philosophy, or more precisely

- philosophical logic, and mathematics. The close connection of modern logic with mathematics makes the question of the mutual relations of these two sciences particularly acute. According to Frege and Russell, mathematics and logic are just two steps in the development of the same science. Mathematics can be completely reduced to logic, and such a purely logical justification of mathematics will make it possible to establish its true and most profound nature.

This approach to the justification of mathematics is called logicism. Modern logic is also closely related to cybernetics - the science of the laws governing processes and systems in any area: in technology, in living organisms, in society. The founder of cybernetics, the American mathematician Wiener, not without reason, emphasized that the very emergence of cybernetics would be unthinkable without mathematical

logic. In addition to cybernetics, modern logic finds wide applications in many other areas of science and technology. Words and things. 1. Language as a sign system. Language is a necessary condition for the existence of abstract thinking. It arose simultaneously with consciousness and thinking. The logical analysis of thinking always takes the form of a study of the language in which it proceeds and without which it is not possible.

In this regard, logic - the science of thinking - is equally the science of language. Language is a system of signs used for the purposes of communication and cognition. The system nature of the language is expressed in the fact that each language, in addition to the dictionary, also has syntax and semantics. The syntactic rules of the language establish ways of forming complex expressions from simple ones. Semantic rules define the ways in which language expressions are given meaning.

Meaning rules are usually divided into three groups: Axiomatic. Such rules require the acceptance of proposals of a certain kind in all circumstances. Deductive. Such rules require the acceptance of the consequences of certain premises if the premises themselves are accepted. Empirical. Such rules of meaning imply transcendence of language and extralinguistic observation. Languages ​​that include empirical rules of meaning are called empirical languages.

All languages ​​can be divided into natural, artificial and partially artificial. 2. Basic functions of the language. The main functions, or use, of the language are those main tasks that are solved by the language in the process of communication and cognition. Among these tasks, a special place is occupied by a description - a message about the real state of affairs. If this message is true, then it is true.

A message that does not correspond to the real state of affairs is false. Another function of language is to try to get something done. Expressions in which the intention of the speaker is realized to ensure that the listener does something are varied. Language can also serve to express a variety of feelings. It can also be used to change the world with a word. "I betroth you" (I declare you husband and wife),

such expressions are called declarations. The declarations do not describe some essential state of affairs. Unlike norms, they are not intended to create a prescribed state of affairs for someone in the future. Declarations directly change the world, and do so by the very fact of their utterance. Language can also be used for communication, that is, in order to impose on the speaker an obligation to perform some future action or adhere to a certain line of behavior.

Language can be used for evaluations, that is, to express a positive, negative or neutral attitude towards the object in question or, if two objects are compared, to express a preference for one of them over the other or to assert their equivalence to each other. Logically, it is important to distinguish between the two main functions of language: descriptive and evaluative. All other uses of the language, apart from psychological and other non-essential

justified from a logical point of view, they are reduced either to descriptions or to assessments. 3. Logical grammar. From grammar, the division of sentences into parts of speech is well known - a noun, an adjective, a verb, etc. The division of language expressions into semantic categories, widely used in logic, resembles this grammatical division and, in principle, originated from it. On this basis, the theory of semantic categories is sometimes called "logical grammar".

Its task is to prevent the confusion of linguistic expressions. different types, which leads to the formation of meaningless expressions. Two expressions are considered to belong to the same semantic category of the language under consideration if the replacement of one of them by another in an arbitrary meaningful sentence does not make this sentence meaningless. Names are language expressions whose substitution in the form "S is P" instead of the variables S and P gives a meaningful sentence.

A sentence (statement) is a linguistic expression that is true or false. A functor is a language expression that is neither a name nor a statement and serves to form new names or statements from existing ones. Names. 1. Types of names. Names - necessary remedy knowledge and communication. Denoting objects and their combinations, names connect the language with the real world.

Names are natural and causal, like the things they are associated with. A name is a language expression denoting a single object, a collection of similar objects, properties, relationships, etc. A language expression is a name if it can be used as the subject “S is P” (S is the subject, P is the predicate). 2. Relationship between names. The content of a name is a set of those properties that are inherent in all objects denoted by data

name, and only them. The scope of a name is a collection, or class, of those objects that have the features included in the content of the name. 3. Definition A definition is a logical operation that reveals the content of a name. To define a name means to indicate what features are included in its content. First of all, it is necessary to note the differences between explicit and implicit definitions. The first have the form of equality - the coincidence of two names (concepts).

Implicit definitions do not take the form of equality between two names. Of particular interest among implicit definitions are contextual and ostensive definitions. Contextual definitions always remain largely incomplete and unstable. Almost all definitions that we encounter in everyday life are contextual definitions. Ostensive definitions are definitions by showing.

Ostensive definitions, like contextual ones, are distinguished by some independence, inconclusiveness. Ostensive definitions - and only they - connect words with things. Without them, language is only a verbal lace, devoid of objective, substantive content. Explicit definitions and, in particular, genus-species definitions are subject to a number of fairly simple and obvious requirements. They are usually called the rules of definition:

Defined and defining concepts should be interchangeable. If one of these concepts occurs in a sentence, it should always be possible to replace it with another. In this case, a sentence that is true before the replacement must remain true after it. To determine through the genus and specific difference, this rule is formulated, as a rule, of the commensurability of the defined and defining concepts: the sets of objects covered by them must be one and the same.

same. One cannot define a name in terms of itself, or define it in terms of such another name, which in turn is defined in terms of it. This rule forbids a vicious circle. The definition must be clear. 4. Division. Division is the operation of distributing into groups those objects that think in the original name. The resulting group division is called the members of the division. The sign by which the division is made is called the basis of the division.

In each division there is, therefore, a divisible concept, the basis of the division, and the members of the division. The requirements for division are quite simple: The division must be carried out on only one basis. This requirement means that an individual attribute or a set of attributes chosen at the beginning as a basis does not follow in the course of division by other attributes.

The division must be commensurate, or exhaustive, that is, the sum of the volumes of the members of the division must be equal to the volume of the concept being divided. This requirement warns against omitting individual members of the division. The members of the division must be mutually exclusive. According to this rule, each individual object must be within the scope of only one visible concept and not be included in the scope of other types of concepts.

The division must be continuous. This rule requires not to make jumps in the division, to move from the original concept to single-order species, but not to subspecies of one of such species. A frequent case of division is dichotomy (literally: division into two). The dichotomy is based on extreme case variation of the attribute, which is the basis of the division: on the one hand, objects that have this attribute are distinguished, on the other hand, they do not have it.

Classification is a multi-stage, branched division. The result of the classification is a system of subordinate names: the divisible name is a genus, new names are species, species of species (subspecies). Statements. 1. Simple and complex statements. Negation, conjunction, disjunction. Sayings - grammatically correct sentence, taken together with the meaning (content) expressed by it

and being true or false. A statement is a more complex formation than a name. When decomposing statements into parts, we always get one or another name. A statement is considered true if the description given by it corresponds to the real situation, and false if it does not correspond to it. "True" and "false" are called the truth-values ​​of the proposition. A statement is called simple if it does not include other statements as its parts.

A statement is complex if it is obtained using logical connectives from several simpler statements. That part of logic, which describes the logical connections of propositions, which does not depend on the structure of simple propositions, is called the general theory of deduction. Negation is a logical connective, with the help of which a new statement is obtained from a given statement, moreover, if the original statement is true, its negation will be false, and vice versa.

The definition of negation can be given the form of a truth table in which "u" means "true" and "l" means "false". A -A I L L I As a result of connecting two statements with the help of the word "and", we get a complex statement called a conjunction. Statements connected in this way are called members of a conjunction. A conjunction is true only if both statements in it are true; if at least one of its terms is false, then the entire conjunction is false.

We denote the conjunction with the symbol &. Truth table for conjunction: a & v and and and and and ll l l l l L L L L L L L L L L L L L L L L L L L with the help of the word "or", we get the dysjunction of these statements. Statements that form a disjunction of these statements are called members of the disjunction. The symbol V will denote a disjunction in the non-exclusive sense, for a disjunction in the exclusive sense, the symbol V` will be used. Tables for two types of disjunction show that non-exclusive disjunction

true when at least one of the statements included in it is true, and false only when both of its members are false; An exclusive disjunction is true when only one of its terms is true, and it is false when both of its terms are true or both are false. А В АВВ AV`В AND AND AND L AND L AND AND L AND AND AND L L L L 2. Conditional statement, implication, equivalence. A conditional statement is a complex statement, usually formulated using the link "if ... then ..." and

establishing that one event, state is in one sense or another the basis or condition for another. A conditional statement is made up of two simple statements. That to which the word "if" is prescribed is called the basis, or antecedent (previous); the statement that comes after the word "that" is called a consequence, or consequential (subsequent). In terms of a conditional statement, the concepts of sufficient and necessary conditions are usually defined;

the antecedent (base) is a sufficient condition for the consequent (consequence), and the consequent is a necessary condition for the antecedent. The conditional statement finds a very wide application in all areas of reasoning. In logic, it is usually represented by means of an implicative statement, or implication. By asserting an implication, we assert that it cannot happen that its reason is true and its consequence false. To establish the truth of the implication "if

A, then B, it suffices to find out the truth-values ​​of A and B. Of the four possible cases, the implication is true in the following three: Both its reason and its consequence are true; The reason is false, but the consequence is true; Both reason and consequence are false. Only in the fourth case, when the reason is true and the consequence false, is the whole implication false. We will denote the implication by the symbol

A B AV I I I I L L L I I L L I Equivalence is more complicated the statement “A if and only if B”, formed from the statements A and B, decomposed into two implications: “if A, then B” and “if B, then A. If logical connectives are defined in terms of true and false, an equivalence is true if and only if both of its constituent statements have the same true value, then

is when both are true or both are false. Let's denote the equivalence by the symbol A B A B I A I I L L L I L L L I MODAL LOGIC 1. LOGICAL MODALITIES Modality is an assessment of an utterance given from one point of view or another. The modal assessment is expressed using the concepts “necessary”, “possible”, “provable”, “refutable”, “mandatory”, “allowed”, etc. Modal statements are statements that contain at least one

from such concepts. Modal statements are divided into types depending on the point of view on the basis of which the characteristics they express are formulated. Modal logic is a section of logic that studies the logical connections of modal statements. Modal logic is composed of a number of sections, or directions, each of which deals with modal statements of a certain type. The foundation of modal logic is propositional logic: the first

there is an extension of the second. The theory of logical modalities studies the connections of logical modal statements, i.e. statements that include logical modal concepts: “logically necessary”, “logically possible”, “logically accidental”, etc. A logically necessary proposition can be defined as a proposition whose negation is a logical contradiction. For example, the statements “It is not true that if neon is an inert gas, then neon is inert

gas" and "It is not true that the grass is green or it is not green." This means that the affirmative statements "If neon is an inert gas then neon is an inert gas" and "The grass is green or it is not green" are logically necessary. The concept of logical necessity is connected with the concept of a logical law: the laws of logic and everything that follows from them are logically necessary. Logically necessary, therefore, all the previously considered

laws of propositional logic. The truth of a logically necessary proposition is established independently of experience, on purely logical grounds. Logical necessity is thus a stronger kind of truth than factual truth. For example, the statement "Snow is white" is in fact true, empirical observation is required to confirm its truth. The statements “Snow is snow”, “White is white”, etc. necessary to be true: to establish

their truth does not need to be appealed to experience, it is enough to know the meanings of the words included in them. Since these statements are logically necessary, each of them can be preceded by the phrase “it is logically necessary that” (“It is logically necessary that snow is snow”, etc.). A logical possibility is the internal consistency of a statement. The statement "The efficiency of a steam engine is 100% is obviously false,

but it is internally consistent and hence logically possible. But the statement “Efficiency such a machine is higher than 100%" is contradictory and therefore logically impossible. A logical possibility can also be defined through the concept of a logical law: a statement is logically possible that does not contradict the laws of logic. Let's say the statement "Microbes are living organisms" is compatible with the laws of logic and, therefore, is logically possible.

The statement “It is not true that if a person is a writer, then he is a writer” contradicts the logical law of identity and therefore is logically impossible. Coincidence is what may or may not be. Chance is not tantamount to a possibility that cannot but be. Randomness is sometimes referred to as "two-way opportunity", i.e. Equal opportunity and statements, and its denial.

A statement is logically accidental when both it and its negation are logically possible. A statement is logically possible that is not self-contradictory. If not only the statement itself, but also its negation do not contain a contradiction, the statement is logically accidental. By chance, for example, the statement "All multicellular beings are mortal": neither the assertion of this fact nor its denial contains an internal (logical) contradiction.

A logically impossible statement is an internally contradictory statement. . For example, statements such as: "Plants breathe and plants do not breathe" and "It is not true that if the Universe is infinite, then it is infinite" are logically impossible. Both of them are denials of logical laws: the first is the law of contradiction, the second is the law of identity. The concepts of logical necessity and possibility can be defined through one another: "A logically necessary" means "negation of

A is not logically possible” (for example: “It is necessary that cold is cold” means “It is impossible that cold should not be cold”); "A is logically possible" means "the negation of A is not logically necessary" ("It is possible that cadmium is a metal" means "It is not true that it is necessary that cadmium is not a metal"). Logical randomness can be defined in terms of logical possibility: "logically random A" means "logically possible both L and not -

A" ("It is logically accidental that there is life on Earth" means "It is logically possible that there is life on Earth, and it is logically possible that there is no life on Earth"). A logically necessary proposition is true, but not vice versa: not every truth is logically necessary. A logically necessary statement is also logically possible, but not vice versa: not everything that is logically possible is logically necessary. From the truth of the proposition follows its logical possibility, but

not vice versa: logical possibility is weaker than truth.

Indicate the period of the traditional stage in the development of logic.

A. 4th century BC. - second half of the 19th century

b. 3rd century BC. - middle of the XIX century.

V. 1st century BC. - beginning of XX century.

Specify a period modern stage development of logic.

A. Mid 19th century - mid-twentieth century

b. Second half of the 19th century - up to our time.

V. Mid 18th century - beginning of XX century.

7. How many premises can there be in a reasoning?

V. One or more.

8. Can the conclusion of one argument become the premise of another?

9. In the correct reasoning of the premises:

V. They can be both true and false statements.

10. In the incorrect reasoning of the premise:

A. Will always be true statements.

b. Will always be false statements.

11. In correct reasoning, the conclusion is:

12. In incorrect reasoning, the conclusion:

A. Will always be a true statement.

b. Will always be false.

V. It can be both true and false.

13. What is a logical fallacy?

A. Violation of the rules and laws of logic.

b. Violation of the rules and laws of human communication.

V. Violation of the rules and laws of thought.

14. What types of logical errors do you know?

A. Sophisms and paralogisms.

b. Sophistry and paradoxes.

V. Paralogisms and paradoxes.

15. The logical form of reasoning is:

A. Its structure, which is revealed as a result of abstraction from the meanings of non-logical terms.

b. Its structure, which is revealed as a result of abstraction from the meanings of logical terms.

3. Decide logical tasks:

Restore the argument in its entirety, that is, identify all its premises and conclusion.

Solution algorithm:

Find premises of reasoning;

Find the conclusion of the reasoning;

Write the premises of the argument one under the other, then write the conclusion of the argument under the line.

Example:

Let's restore the reasoning of the ancient Roman philosopher Lucretius Cyrus: "What changes, collapses, and, therefore, perishes."

1. That which changes is destroyed.

2. That which is destroyed perishes.

_______________________________________

Therefore, what changes, perishes.

Tasks

1.1. Bringing a person to suicide is a crime against life. Ivanov committed a crime against life.

1.2. “The army with which the sovereign defends his country is either his own, or allied, or hired, or mixed. Mercenary and allied troops are useless and dangerous” (Machiavelli).

1.3. If the premises are true and the reasoning is correct, then the conclusion is true. Therefore, reasoning is wrong if the premises are not true propositions.

1.4. The conference was a success, hence it was well organized.

1.5. "Cogito, ergo sum" ("I think, therefore I am").

Determine the logical form of statements.

Solution algorithm:

In order to complete this task, you must:

Define the logical terms that make up the statement;

Identify simple statements (non-logical terms) that make up the statement. Designate them with certain signs;

Write down the logical form of the statement.

Example:

Consider the statement: "If I study for an exam, I will pass it."

This expression contains one logical term: "if...then...".

It consists of two simple statements:

1. I will prepare for the exam - p.

2. I will successfully pass the exam - q.

The logical form of the statement: "If p, then q."

Tasks

2.1. Logic is a science or an art.

2.2. If logic is a science, then it is not an art.

2.3. Logic is both a science and an art.

2.4. If he goes on vacation in the summer, he will go on vacation to Turkey or Cyprus.

2.5. When you admit own mistakes, you have a chance to fix them and not allow them anymore.

Determine the logical form of reasoning.

Solution algorithm:

In order to complete this task, you must:

Find the premises and conclusion of the argument. If the reasoning is not given in full, then restore it;

Define the logical terms that make up the premises and conclusion of the reasoning;

Replace simple statements that are part of the premises and conclusion of the argument, and designate them with certain signs;

Write down the logical form of reasoning.

Example:

Consider the reasoning of Augustine the Blessed: "If one of the elect perishes, then God is mistaken, but none of the elect perish, for God does not make mistakes."

Find the premises and conclusion of the argument.

1. If one of the elect dies, then God is mistaken.

2. God doesn't make mistakes.

_________________________________________

Therefore, none of the chosen ones perish.

The composition of the first premise includes the logical term "if ..., then ...", the second premise and conclusion - the logical term "not".

The premises and conclusion of the argument consist of two simple statements:

One of the chosen ones dies.

God is wrong.

Let's designate them respectively р, q.

Let's write a logical form of reasoning.

If p, then q.

__________________

Therefore, not r.

3.1. This court decision is not acquittal, because it involves dismissal from work.

3.2. If an action is required, then it is not prohibited. That is not forbidden, it is allowed. Therefore, if an action is required, then it is allowed.

3.3. “If death is a transition into non-existence, then it is good. If death is a transition to another world, then it is a blessing. Death is a transition to non-existence or to another world. Therefore, death is good” (Socrates).

3.4. “If capital investment remains constant, government spending will increase or unemployment will arise. If government spending does not increase, then taxes will be reduced. If taxes are lowered and capital investment remains constant, then unemployment will not increase. Consequently, government spending will increase.”

3.5. If Peter goes to Moscow, then Ivan will go to Samara. Peter will go to Moscow or St. Petersburg. If Peter goes to St. Petersburg, then Anna will stay in Arkhangelsk. Consequently, Ivan will go to Samara or Saratov.

In correct reasoning, the conclusion follows from the premises with logical necessity, and the general scheme of such reasoning is a logical law.

Logical laws thus underlie logically perfect thinking. To reason logically correctly means to reason in accordance with the laws of logic.

There are an infinite number of schemes for correct reasoning (logical laws). Many of them are known to us from the practice of reasoning. We apply them intuitively, without realizing that in each correctly drawn conclusion one or another logical law is used.

Here are some of the most commonly used schemes.

“If there is a first, then there is a second; there is the first; therefore, there is a second. This scheme allows us to pass from the statement of the conditional statement and the statement of its foundation to the statement of the consequence. According to this scheme, in particular, the reasoning proceeds: “If ice is heated, it melts; ice is heated; it means he's melting."

This logically correct movement of thought is sometimes confused with its similar but logically incorrect movement from asserting the consequence of a conditional statement to asserting its foundation: “If there is a first, then there is a second; so there is a first. The last scheme is not a logical law; from true premises, it can lead to a false conclusion.

For example, the reasoning following this scheme “If a person is eighty years old, he is old; the person is old; therefore the man is eighty years old" leads to the erroneous conclusion that the old man is exactly eighty years old.

“If there is a first, then there is a second; but there is no second; means no first. By means of this scheme, from the affirmation of a conditional statement and the negation of its consequence, a transition is made to the negation of the foundation of the statement.

For example: “If the day comes, it becomes light; but now it is not light; therefore the day has not come.”

Sometimes this scheme is confused with the logically incorrect movement of thought from denying the basis of the conditional statement to denying its consequence: “If there is the first, there is the second; but the first is not; so there is no second.

For example: If a person has a fever, he is sick; but he does not have a fever; It means he's not sick.

“If there is a first, then there is a second; therefore, if there is no second, then there is no first". This scheme allows, using negation, to swap statements.

For example, from the statement "If there is thunder, there is also lightning", the statement "If there is no lightning, then there is no thunder" is obtained.

“There is at least either the first or the second; on the first no; so there is a second».

For example: “It happens day or night: now there is no night; therefore, it is now day."

“Either the first or the second takes place; there is a first, so there is no second". Through this scheme, from the assertion of two mutually exclusive alternatives and the establishment of which of them is available, the transition is made to the denial of the other alternative:

For example: 1. “Dostoevsky was born either in Moscow or in St. Petersburg; he was born in Moscow; so it is not true that he was born in St. Petersburg.

2. The American western The Good, the Bad and the Ugly talks about this division of human roles. The bandit says: “Remember, One-armed, that the world is divided into two parts: those who hold a revolver, V those who dig. I have the revolver now.’ so grab a shovel.” This reasoning is also based on the scheme under consideration.

« It is not true that there is both the first and the second; therefore, there is no first or there is no second "", "There is a first or there is a second; therefore, it is not true that there is no first and no second *. These and similar schemes allow you to move from statements with the union “and” to statements with the union “or”, and vice versa.

For example: Using these diagrams, one can go from the statement “It is not true that it is windy and raining today” to the statement “It is not true that it is windy today, or it is not true that it is raining today” and from the statement “Amundsen or Scott was the first at the South Pole” to "It is not true that neither Amundsen nor Scott is the first person to visit the South Pole."

These are some of the infinity of patterns of correct reasoning. In the future, these and other schemes will be considered in more detail and presented using special logical symbols.